Maximal Distortion of Geodesic Diameters in Polygonal Domains
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For a polygon P with holes in the plane, we denote by ϱ(P) the ratio between the geodesic and the Euclidean diameters of P. It is shown that over all convex polygons with h convex holes, the supremum of ϱ(P) is between Ω(h^1/3) and O(h^1/2). The upper bound improves to O(1+min{h^3/4Δ,h^1/2Δ^1/2}) if every hole has diameter at most Δ· diam_2(P); and to O(1) if every hole is a fat convex polygon. Furthermore, we show that the function g(h)=sup_P ϱ(P) over convex polygons with h convex holes has the same growth rate as an analogous quantity over geometric triangulations with h vertices when h→∞.
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